In mathematics, differential topology is the field dealing with differentiable functions on differentiable manifolds. It is closely related to differential geometry and together they make up the geometric theory of differentiable manifolds.
Differential geometry is a mathematical discipline that uses the techniques of differential calculus, integral calculus, linear algebra and multilinear algebra to study problems in geometry. The theory of plane and space curves and surfaces in the three-dimensional Euclidean space formed the basis for development of differential geometry during the 18th century and the 19th century.
In mathematics, the Poincaré conjecture is a theorem about the characterization of the 3-sphere, which is the hypersphere that bounds the unit ball in four-dimensional space. The conjecture states:
Every simply connected, closed 3-manifold is homeomorphic to the 3-sphere.
In differential geometry, a (smooth) Riemannian manifold or (smooth) Riemannian space(M, g) is a real, smooth manifold M equipped with an inner product gp on the tangent space TpM at each point p that varies smoothly from point to point in the sense that if X and Y are differentiable vector fields on M, then p ↦ gp(X|p, Y|p) is a smooth function. The family gp of inner products is called a Riemannian metric. These terms are named after the German mathematician Bernhard Riemann. The study of Riemannian manifolds constitutes the subject called Riemannian geometry.
In algebraic geometry, a Calabi–Yau manifold, also known as a Calabi–Yau space, is a particular type of manifold which has properties, such as Ricci flatness, yielding applications in theoretical physics. Particularly in superstring theory, the extra dimensions of spacetime are sometimes conjectured to take the form of a 6-dimensional Calabi–Yau manifold, which led to the idea of mirror symmetry. Their name was coined by Candelas et al. (1985), after Eugenio Calabi who first conjectured that such surfaces might exist, and Shing-Tung Yau (1978) who proved the Calabi conjecture.
Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, smooth manifolds with a Riemannian metric, i.e. with an inner product on the tangent space at each point that varies smoothly from point to point. This gives, in particular, local notions of angle, length of curves, surface area and volume. From those, some other global quantities can be derived by integrating local contributions.
In differential geometry, the Ricci curvature tensor, named after Gregorio Ricci-Curbastro, represents the amount by which the volume of a narrow conical piece of a small geodesic ball in a curved Riemannian manifold deviates from that of the standard ball in Euclidean space. As such, it provides one way of measuring the degree to which the geometry determined by a given Riemannian metric might differ from that of ordinary Euclidean n-space. The Ricci tensor is defined on any pseudo-Riemannian manifold, as a trace of the Riemann curvature tensor. Like the metric itself, the Ricci tensor is a symmetric bilinear form on the tangent space of the manifold.
In differential geometry, a pseudo-Riemannian manifold, also called a semi-Riemannian manifold, is a differentiable manifold with a metric tensor that is everywhere nondegenerate. This is a generalization of a Riemannian manifold in which the requirement of positive-definiteness is relaxed.
In mathematics and especially differential geometry, a Kähler manifold is a manifold with three mutually compatible structures: a complex structure, a Riemannian structure, and a symplectic structure. The concept was first studied by Jan Arnoldus Schouten and David van Dantzig in 1930, and then introduced by Erich Kähler in 1933. The terminology has been fixed by André Weil.
In differential geometry, a complex manifold is a manifold with an atlas of charts to the open unit disk in Cn, such that the transition maps are holomorphic.
In mathematics, geometric topology is the study of manifolds and maps between them, particularly embeddings of one manifold into another.
In automotive engineering, an inlet manifold or intake manifold is the part of an engine that supplies the fuel/air mixture to the cylinders. The word manifold comes from the Old English word manigfeald and refers to the multiplying of one (pipe) into many.
A manifold is an abstract mathematical space which, in a close-up view, resembles the spaces described by Euclidean geometry.
Manifold vacuum, or engine vacuum in an internal combustion engine is the difference in air pressure between the engine's intake manifold and Earth's atmosphere.
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, each point of an n-dimensional manifold has a neighbourhood that is homeomorphic to the Euclidean space of dimension n. In this more precise terminology, a manifold is referred to as an n-manifold.
In topology, a branch of mathematics, a topological manifold is a topological space which locally resembles real n-dimensional space in a sense defined below. Topological manifolds form an important class of topological spaces with applications throughout mathematics. All manifolds are topological manifolds by definition, but many manifolds may be equipped with additional structure. When the phrase "topological manifold" is used, it is usually done to emphasize that the manifold does not have any additional structure, or that only the "underlying" topological manifold is being considered. Every manifold has an "underlying" topological manifold, gotten by simply "forgetting" any additional structure the manifold has.
In mathematics, a differentiable manifold is a type of manifold that is locally similar enough to a linear space to allow one to do calculus. Any manifold can be described by a collection of charts, also known as an atlas. One may then apply ideas from calculus while working within the individual charts, since each chart lies within a linear space to which the usual rules of calculus apply. If the charts are suitably compatible, then computations done in one chart are valid in any other differentiable chart.
David Gabai, a mathematician, is the Hughes-Rogers Professor of Mathematics at Princeton University. Focused on low-dimensional topology and hyperbolic geometry, he is a leading researcher in those subjects.
Charles L. Epstein is a Thomas A. Scott Professor of Mathematics at the University of Pennsylvania, Philadelphia.
William Philip Minicozzi II is an American mathematician. He was born in Bryn Mawr, Pennsylvania, in 1967.
